Validity of the splitting and full-separation concepts

When Fourier transformation is possible, extrapolation operators are complex numbers like e^{i k_z z}. With complex numbers a and b there is never any question that $ab\ =\ ba$.Then both splitting and full separation are always valid, but the proof will be given only for a more general arrangement.

Suppose Fourier transformation has not been done, or could not be done because of some spatial variation of material properties. Then extrapolation operators are built up by combinations of the finite-differencing operators described in previous sections. Let A and B denote two such operators. For example, A could be a matrix containing the second x differencing operator. Seen as matrices, the boundary conditions of a differential operator are incorporated in the corners of the matrix. The bottom line is whether AB= BA, so the question clearly involves the boundary conditions as well as the differential operators.

Extrapolation forward a short distance can be done with the operator $( I+ A\, \Delta z )$.In two-dimensional problems A was seen to be a four-dimensional matrix. For convenience the terms of the four-dimensional matrix can be arranged into a super-large, ordinary two-dimensional matrix. Implicit finite-differencing calculations gave extrapolation operators like $( I+ A\, \Delta z )/( I- A\, \Delta z )$.Let ${\bf p}$ denote a vector where components of the vector designate the wavefield at various locations. As has been seen, the locations need not be constrained to the x-axis but could also be distributed throughout the (x,y)-plane. Numerical analysis gives us a matrix operator, say A, which enables us to project forward, say,

$${\bf p} (z + \, \Delta z) \eq A_1 \ {\bf p} (z)$$ (14)

The subscript on A denotes the fact that the operator may change with z. To get a step further the operator is applied again, say,

$${\bf p} (z + 2 \, \Delta z) \eq A_2 \ [ A_1 \ {\bf p} (z)]$$ (15)

From an operational point of view the matrix A is never squared, but from an analytical point of view, it really is squared.

$$A_2 \ [ A_1 \ {\bf p} (z)] \eq ( A_2 \ A_1 ) \ {\bf p} (z)$$ (16)

To march some distance down the z-axis we apply the operator many times. Take an interval $z_1\ -\ z_0$, to be divided into N subintervals. Since there are N intervals, an error proportional to 1/N in each subinterval would accumulate to an unacceptable level by the time z₁ was reached. On the other hand, an error proportional to 1 / N² could only accumulate to a total error proportional to 1/N. Such an error would disappear as the number of subintervals increased.

To prove the validity of splitting, we take $\Delta z\ =\ ( z_1\ -\ z_0 ) / N$.Observe that the operator $I+ ( A+ B) \Delta z$ differs from the operator $( I+ A\, \Delta z)( I+ B\, \Delta z)$ by something in proportion to $\Delta z^2$ or 1/N². So in the limit of a very large number of subintervals, the error disappears.

It is much easier to establish the validity of the full-separation concept. Commutativity is whether or not $A\, B\, = \, B\, A$.Commutativity is always true for scalars. With finite differencing the question is whether the two matrices commute. Taking A and B to be differential operators, commutativity is defined with the help of the family of all possible wavefields P. Then A and B are commutative if $A\, B\, P \, = \, B\, A\, P$.

The operator representing $\partial P / \partial z$ will be taken to be A+ B. The simplest numerical integration scheme using the splitting method is  

$$P( z_0\ +\ \Delta z)\eq ( I\ +\ A\, \Delta z )\ ( I\ +\ B\, \Delta z ) \ P ( z_0 )$$ (17)

Applying (17) in many stages gives a product of many operators. The operators A and B are subscripted with j to denote the possibility that they change with z.  

$$P ( z_1 ) \eq \prod_{j=1}^N\ [ ( I\ +\ A_j \, \Delta z ) ( I\ +\ B_j \, \Delta z ) ] \ P ( z_0 )$$ (18)

As soon as A and B are assumed to be commutative, the factors in (18) may be rearranged at will. For example, the A operator could be applied in its entirety before the B operator is applied:  

$$P ( z_1 ) \eq \left[ \prod_{j=1}^N\ ( I\ +\ B_j \, \Delta z ... ...left[ \prod_{j=1}^N\ ( I\ +\ A_j \, \Delta z) \right] P ( z_0 )$$ (19)

Thus the full-separation concept is seen to depend on the commutativity of operators.

EXERCISES:

1. With a splitting method, Ma Zaitian (Ma [1981]) showed how very wide-angle representations may be implemented with successive applications of an equation like a 45$^\circ$ equation. This avoids the band matrix solving inherent in the high-order Muir expansion. Specifically, one chooses coefficients a_j and b_j, in the square-root fitting function

   $$i \, k_z \ \eq \ \sum_{j=1}^{n-1} \ \ { k_x^2 \ b_j \over -i \omega \ +\ a_j \, i k_x }$$ (20)

   The general $n^{{\rm th}}$-order case is somewhat complicated, so your job is simply to find a₁, a₂, b₁, and b₂, to make the fitting function match the 45$^\circ$ equation.
2. Migrate a two dimensional data set with velocity v₁. Then migrate the migrated data set with a velocity v₂. Rocca pointed out that this double migration simulates a migration with a third velocity v₃. Using a method of deduction similar to the Jakubowicz deduction equations (10), (11), and (12) find v₃ in terms of v₁ and v₂.
3. Consider migration of zero-offset data P(x,y,t) recorded in an area of the earth's surface plane. Assume a computer with a random access memory (RAM) large enough to hold several planes (any orientation) from the data volume. (The entire volume resides in slow memory devices). Define a migration algorithm by means of a program sketch. Your method should allow velocity to vary with depth.